Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E

Percentage Accurate: 99.7% → 99.8%

Time: 5.2s

Alternatives: 6

Speedup: 1.3×

• 24.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 * (2.0d0 - (x * 3.0d0))) * x
end function

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Initial Program: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (* 3.0 (- 2.0 (* x 3.0))) x))

C

double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (3.0d0 * (2.0d0 - (x * 3.0d0))) * x
end function

Java

public static double code(double x) {
	return (3.0 * (2.0 - (x * 3.0))) * x;
}

Python

def code(x):
	return (3.0 * (2.0 - (x * 3.0))) * x

Julia

function code(x)
	return Float64(Float64(3.0 * Float64(2.0 - Float64(x * 3.0))) * x)
end

MATLAB

function tmp = code(x)
	tmp = (3.0 * (2.0 - (x * 3.0))) * x;
end

Wolfram

code[x_] := N[(N[(3.0 * N[(2.0 - N[(x * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x 6.0 (* -9.0 (pow x 2.0))))

C

double code(double x) {
	return fma(x, 6.0, (-9.0 * pow(x, 2.0)));
}

Julia

function code(x)
	return fma(x, 6.0, Float64(-9.0 * (x ^ 2.0)))
end

Wolfram

code[x_] := N[(x * 6.0 + N[(-9.0 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

   2. cancel-sign-sub-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

   3. distribute-rgt-in 99.7%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

   4. metadata-eval 99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

   5. distribute-lft-neg-out 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

   7. associate-*l* 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

   9. metadata-eval 99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 99.7%

      \[\leadsto \color{blue}{6 \cdot x + \left(x \cdot -9\right) \cdot x} \]

   2. *-commutative 99.7%

      \[\leadsto \color{blue}{x \cdot 6} + \left(x \cdot -9\right) \cdot x \]

   3. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, \left(x \cdot -9\right) \cdot x\right)} \]

   4. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{\left(-9 \cdot x\right)} \cdot x\right) \]

   5. associate-*l* 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, \color{blue}{-9 \cdot \left(x \cdot x\right)}\right) \]

   6. pow2 99.8%

      \[\leadsto \mathsf{fma}\left(x, 6, -9 \cdot \color{blue}{{x}^{2}}\right) \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right)} \]

6. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(x, 6, -9 \cdot {x}^{2}\right) \]

Alternative 2: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 0.66\right):\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.68) (not (<= x 0.66))) (* -9.0 (* x x)) (* x 6.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.68) || !(x <= 0.66)) {
		tmp = -9.0 * (x * x);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-0.68d0)) .or. (.not. (x <= 0.66d0))) then
        tmp = (-9.0d0) * (x * x)
    else
        tmp = x * 6.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -0.68) || !(x <= 0.66)) {
		tmp = -9.0 * (x * x);
	} else {
		tmp = x * 6.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -0.68) or not (x <= 0.66):
		tmp = -9.0 * (x * x)
	else:
		tmp = x * 6.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.68) || !(x <= 0.66))
		tmp = Float64(-9.0 * Float64(x * x));
	else
		tmp = Float64(x * 6.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -0.68) || ~((x <= 0.66)))
		tmp = -9.0 * (x * x);
	else
		tmp = x * 6.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.68], N[Not[LessEqual[x, 0.66]], $MachinePrecision]], N[(-9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], N[(x * 6.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.680000000000000049 or 0.660000000000000031 < x

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.8%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Step-by-step derivation

      1. flip-+ 99.7%

         \[\leadsto x \cdot \color{blue}{\frac{6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}{6 - x \cdot -9}} \]

      2. associate-*r/ 81.7%

         \[\leadsto \color{blue}{\frac{x \cdot \left(6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right)}{6 - x \cdot -9}} \]

      3. metadata-eval 81.7%

         \[\leadsto \frac{x \cdot \left(\color{blue}{36} - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right)}{6 - x \cdot -9} \]

      4. swap-sqr 81.6%

         \[\leadsto \frac{x \cdot \left(36 - \color{blue}{\left(x \cdot x\right) \cdot \left(-9 \cdot -9\right)}\right)}{6 - x \cdot -9} \]

      5. pow2 81.6%

         \[\leadsto \frac{x \cdot \left(36 - \color{blue}{{x}^{2}} \cdot \left(-9 \cdot -9\right)\right)}{6 - x \cdot -9} \]

      6. metadata-eval 81.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot \color{blue}{81}\right)}{6 - x \cdot -9} \]

      7. *-commutative 81.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 - \color{blue}{-9 \cdot x}} \]

      8. cancel-sign-sub-inv 81.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{\color{blue}{6 + \left(--9\right) \cdot x}} \]

      9. metadata-eval 81.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + \color{blue}{9} \cdot x} \]

   5. Applied egg-rr 81.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + 9 \cdot x}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\frac{x}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{x}{\frac{6 + \color{blue}{x \cdot 9}}{36 - {x}^{2} \cdot 81}} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{6 + x \cdot 9}{36 - {x}^{2} \cdot 81}}} \]

   8. Taylor expanded in x around inf 98.3%

      \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 98.3%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{-0.1111111111111111}{x}} \]

      2. div-inv 98.2%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{-0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. times-frac 98.3%

         \[\leadsto \color{blue}{\frac{1}{-0.1111111111111111} \cdot \frac{x}{\frac{1}{x}}} \]

      4. metadata-eval 98.3%

         \[\leadsto \color{blue}{-9} \cdot \frac{x}{\frac{1}{x}} \]

   10. Applied egg-rr 98.3%

       \[\leadsto \color{blue}{-9 \cdot \frac{x}{\frac{1}{x}}} \]
   11. Step-by-step derivation

       1. pow1 98.3%

          \[\leadsto -9 \cdot \frac{\color{blue}{{x}^{1}}}{\frac{1}{x}} \]

       2. inv-pow 98.3%

          \[\leadsto -9 \cdot \frac{{x}^{1}}{\color{blue}{{x}^{-1}}} \]

       3. pow-div 98.3%

          \[\leadsto -9 \cdot \color{blue}{{x}^{\left(1 - -1\right)}} \]

       4. metadata-eval 98.3%

          \[\leadsto -9 \cdot {x}^{\color{blue}{2}} \]

       5. unpow2 98.3%

          \[\leadsto -9 \cdot \color{blue}{\left(x \cdot x\right)} \]

   12. Applied egg-rr 98.3%

       \[\leadsto -9 \cdot \color{blue}{\left(x \cdot x\right)} \]

   if -0.680000000000000049 < x < 0.660000000000000031

   1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

      2. cancel-sign-sub-inv 99.6%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

   4. Taylor expanded in x around 0 98.3%

      \[\leadsto x \cdot \color{blue}{6} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68 \lor \neg \left(x \leq 0.66\right):\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 6\\ \end{array} \]

Alternative 3: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.68)
   (* -9.0 (* x x))
   (if (<= x 0.66) (* x 6.0) (* x (* x -9.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.68) {
		tmp = -9.0 * (x * x);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.68d0)) then
        tmp = (-9.0d0) * (x * x)
    else if (x <= 0.66d0) then
        tmp = x * 6.0d0
    else
        tmp = x * (x * (-9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.68) {
		tmp = -9.0 * (x * x);
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.68:
		tmp = -9.0 * (x * x)
	elif x <= 0.66:
		tmp = x * 6.0
	else:
		tmp = x * (x * -9.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.68)
		tmp = Float64(-9.0 * Float64(x * x));
	elseif (x <= 0.66)
		tmp = Float64(x * 6.0);
	else
		tmp = Float64(x * Float64(x * -9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.68)
		tmp = -9.0 * (x * x);
	elseif (x <= 0.66)
		tmp = x * 6.0;
	else
		tmp = x * (x * -9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.68], N[(-9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.66], N[(x * 6.0), $MachinePrecision], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.680000000000000049

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]
   4. Step-by-step derivation

      1. flip-+ 99.6%

         \[\leadsto x \cdot \color{blue}{\frac{6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)}{6 - x \cdot -9}} \]

      2. associate-*r/ 84.7%

         \[\leadsto \color{blue}{\frac{x \cdot \left(6 \cdot 6 - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right)}{6 - x \cdot -9}} \]

      3. metadata-eval 84.7%

         \[\leadsto \frac{x \cdot \left(\color{blue}{36} - \left(x \cdot -9\right) \cdot \left(x \cdot -9\right)\right)}{6 - x \cdot -9} \]

      4. swap-sqr 84.6%

         \[\leadsto \frac{x \cdot \left(36 - \color{blue}{\left(x \cdot x\right) \cdot \left(-9 \cdot -9\right)}\right)}{6 - x \cdot -9} \]

      5. pow2 84.6%

         \[\leadsto \frac{x \cdot \left(36 - \color{blue}{{x}^{2}} \cdot \left(-9 \cdot -9\right)\right)}{6 - x \cdot -9} \]

      6. metadata-eval 84.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot \color{blue}{81}\right)}{6 - x \cdot -9} \]

      7. *-commutative 84.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 - \color{blue}{-9 \cdot x}} \]

      8. cancel-sign-sub-inv 84.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{\color{blue}{6 + \left(--9\right) \cdot x}} \]

      9. metadata-eval 84.6%

         \[\leadsto \frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + \color{blue}{9} \cdot x} \]

   5. Applied egg-rr 84.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left(36 - {x}^{2} \cdot 81\right)}{6 + 9 \cdot x}} \]
   6. Step-by-step derivation

      1. associate-/l* 99.6%

         \[\leadsto \color{blue}{\frac{x}{\frac{6 + 9 \cdot x}{36 - {x}^{2} \cdot 81}}} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{x}{\frac{6 + \color{blue}{x \cdot 9}}{36 - {x}^{2} \cdot 81}} \]

   7. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{x}{\frac{6 + x \cdot 9}{36 - {x}^{2} \cdot 81}}} \]

   8. Taylor expanded in x around inf 98.0%

      \[\leadsto \frac{x}{\color{blue}{\frac{-0.1111111111111111}{x}}} \]
   9. Step-by-step derivation

      1. *-un-lft-identity 98.0%

         \[\leadsto \frac{\color{blue}{1 \cdot x}}{\frac{-0.1111111111111111}{x}} \]

      2. div-inv 98.0%

         \[\leadsto \frac{1 \cdot x}{\color{blue}{-0.1111111111111111 \cdot \frac{1}{x}}} \]

      3. times-frac 98.1%

         \[\leadsto \color{blue}{\frac{1}{-0.1111111111111111} \cdot \frac{x}{\frac{1}{x}}} \]

      4. metadata-eval 98.1%

         \[\leadsto \color{blue}{-9} \cdot \frac{x}{\frac{1}{x}} \]

   10. Applied egg-rr 98.1%

       \[\leadsto \color{blue}{-9 \cdot \frac{x}{\frac{1}{x}}} \]
   11. Step-by-step derivation

       1. pow1 98.1%

          \[\leadsto -9 \cdot \frac{\color{blue}{{x}^{1}}}{\frac{1}{x}} \]

       2. inv-pow 98.1%

          \[\leadsto -9 \cdot \frac{{x}^{1}}{\color{blue}{{x}^{-1}}} \]

       3. pow-div 98.1%

          \[\leadsto -9 \cdot \color{blue}{{x}^{\left(1 - -1\right)}} \]

       4. metadata-eval 98.1%

          \[\leadsto -9 \cdot {x}^{\color{blue}{2}} \]

       5. unpow2 98.1%

          \[\leadsto -9 \cdot \color{blue}{\left(x \cdot x\right)} \]

   12. Applied egg-rr 98.1%

       \[\leadsto -9 \cdot \color{blue}{\left(x \cdot x\right)} \]

   if -0.680000000000000049 < x < 0.660000000000000031

   1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

      2. cancel-sign-sub-inv 99.6%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

   4. Taylor expanded in x around 0 98.3%

      \[\leadsto x \cdot \color{blue}{6} \]

   if 0.660000000000000031 < x

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.8%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto x \cdot \color{blue}{\left(-9 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   6. Simplified 98.7%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;-9 \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \end{array} \]

Alternative 4: 97.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;3 \cdot \left(x \cdot \left(x \cdot -3\right)\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.68)
   (* 3.0 (* x (* x -3.0)))
   (if (<= x 0.66) (* x 6.0) (* x (* x -9.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.68) {
		tmp = 3.0 * (x * (x * -3.0));
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= (-0.68d0)) then
        tmp = 3.0d0 * (x * (x * (-3.0d0)))
    else if (x <= 0.66d0) then
        tmp = x * 6.0d0
    else
        tmp = x * (x * (-9.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= -0.68) {
		tmp = 3.0 * (x * (x * -3.0));
	} else if (x <= 0.66) {
		tmp = x * 6.0;
	} else {
		tmp = x * (x * -9.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -0.68:
		tmp = 3.0 * (x * (x * -3.0))
	elif x <= 0.66:
		tmp = x * 6.0
	else:
		tmp = x * (x * -9.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.68)
		tmp = Float64(3.0 * Float64(x * Float64(x * -3.0)));
	elseif (x <= 0.66)
		tmp = Float64(x * 6.0);
	else
		tmp = Float64(x * Float64(x * -9.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -0.68)
		tmp = 3.0 * (x * (x * -3.0));
	elseif (x <= 0.66)
		tmp = x * 6.0;
	else
		tmp = x * (x * -9.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -0.68], N[(3.0 * N[(x * N[(x * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 0.66], N[(x * 6.0), $MachinePrecision], N[(x * N[(x * -9.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.680000000000000049

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. associate-*l* 99.9%

         \[\leadsto \color{blue}{3 \cdot \left(\left(2 - x \cdot 3\right) \cdot x\right)} \]

      2. *-commutative 99.9%

         \[\leadsto 3 \cdot \left(\left(2 - \color{blue}{3 \cdot x}\right) \cdot x\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{3 \cdot \left(\left(2 - 3 \cdot x\right) \cdot x\right)} \]

   4. Taylor expanded in x around inf 98.1%

      \[\leadsto 3 \cdot \left(\color{blue}{\left(-3 \cdot x\right)} \cdot x\right) \]
   5. Step-by-step derivation

      1. *-commutative 98.1%

         \[\leadsto 3 \cdot \left(\color{blue}{\left(x \cdot -3\right)} \cdot x\right) \]

   6. Simplified 98.1%

      \[\leadsto 3 \cdot \left(\color{blue}{\left(x \cdot -3\right)} \cdot x\right) \]

   if -0.680000000000000049 < x < 0.660000000000000031

   1. Initial program 99.6%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

      2. cancel-sign-sub-inv 99.6%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.7%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

   4. Taylor expanded in x around 0 98.3%

      \[\leadsto x \cdot \color{blue}{6} \]

   if 0.660000000000000031 < x

   1. Initial program 99.7%

      \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
   2. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

      2. cancel-sign-sub-inv 99.7%

         \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

      3. distribute-rgt-in 99.7%

         \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

      4. metadata-eval 99.7%

         \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

      5. distribute-lft-neg-out 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

      6. distribute-rgt-neg-in 99.7%

         \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

      7. associate-*l* 99.8%

         \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

      8. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

      9. metadata-eval 99.8%

         \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

   4. Taylor expanded in x around inf 98.7%

      \[\leadsto x \cdot \color{blue}{\left(-9 \cdot x\right)} \]
   5. Step-by-step derivation

      1. *-commutative 98.7%

         \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]

   6. Simplified 98.7%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot -9\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.68:\\ \;\;\;\;3 \cdot \left(x \cdot \left(x \cdot -3\right)\right)\\ \mathbf{elif}\;x \leq 0.66:\\ \;\;\;\;x \cdot 6\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot -9\right)\\ \end{array} \]

Alternative 5: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(6 + x \cdot -9\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 6.0 (* x -9.0))))

C

double code(double x) {
	return x * (6.0 + (x * -9.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (6.0d0 + (x * (-9.0d0)))
end function

Java

public static double code(double x) {
	return x * (6.0 + (x * -9.0));
}

Python

def code(x):
	return x * (6.0 + (x * -9.0))

Julia

function code(x)
	return Float64(x * Float64(6.0 + Float64(x * -9.0)))
end

MATLAB

function tmp = code(x)
	tmp = x * (6.0 + (x * -9.0));
end

Wolfram

code[x_] := N[(x * N[(6.0 + N[(x * -9.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

   2. cancel-sign-sub-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

   3. distribute-rgt-in 99.7%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

   4. metadata-eval 99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

   5. distribute-lft-neg-out 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

   7. associate-*l* 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

   9. metadata-eval 99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

4. Final simplification 99.7%

   \[\leadsto x \cdot \left(6 + x \cdot -9\right) \]

Alternative 6: 50.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ x \cdot 6 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 6.0))

C

double code(double x) {
	return x * 6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 6.0d0
end function

Java

public static double code(double x) {
	return x * 6.0;
}

Python

def code(x):
	return x * 6.0

Julia

function code(x)
	return Float64(x * 6.0)
end

MATLAB

function tmp = code(x)
	tmp = x * 6.0;
end

Wolfram

code[x_] := N[(x * 6.0), $MachinePrecision]

Derivation

1. Initial program 99.7%

   \[\left(3 \cdot \left(2 - x \cdot 3\right)\right) \cdot x \]
2. Step-by-step derivation

   1. *-commutative 99.7%

      \[\leadsto \color{blue}{x \cdot \left(3 \cdot \left(2 - x \cdot 3\right)\right)} \]

   2. cancel-sign-sub-inv 99.7%

      \[\leadsto x \cdot \left(3 \cdot \color{blue}{\left(2 + \left(-x\right) \cdot 3\right)}\right) \]

   3. distribute-rgt-in 99.7%

      \[\leadsto x \cdot \color{blue}{\left(2 \cdot 3 + \left(\left(-x\right) \cdot 3\right) \cdot 3\right)} \]

   4. metadata-eval 99.7%

      \[\leadsto x \cdot \left(\color{blue}{6} + \left(\left(-x\right) \cdot 3\right) \cdot 3\right) \]

   5. distribute-lft-neg-out 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(-x \cdot 3\right)} \cdot 3\right) \]

   6. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{\left(x \cdot \left(-3\right)\right)} \cdot 3\right) \]

   7. associate-*l* 99.7%

      \[\leadsto x \cdot \left(6 + \color{blue}{x \cdot \left(\left(-3\right) \cdot 3\right)}\right) \]

   8. metadata-eval 99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \left(\color{blue}{-3} \cdot 3\right)\right) \]

   9. metadata-eval 99.7%

      \[\leadsto x \cdot \left(6 + x \cdot \color{blue}{-9}\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \left(6 + x \cdot -9\right)} \]

4. Taylor expanded in x around 0 49.7%

   \[\leadsto x \cdot \color{blue}{6} \]

5. Final simplification 49.7%

   \[\leadsto x \cdot 6 \]

Developer target: 99.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 6 \cdot x - 9 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (* 6.0 x) (* 9.0 (* x x))))

C

double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (6.0d0 * x) - (9.0d0 * (x * x))
end function

Java

public static double code(double x) {
	return (6.0 * x) - (9.0 * (x * x));
}

Python

def code(x):
	return (6.0 * x) - (9.0 * (x * x))

Julia

function code(x)
	return Float64(Float64(6.0 * x) - Float64(9.0 * Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = (6.0 * x) - (9.0 * (x * x));
end

Wolfram

code[x_] := N[(N[(6.0 * x), $MachinePrecision] - N[(9.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023305 
(FPCore (x)
  :name "Diagrams.Tangent:$catParam from diagrams-lib-1.3.0.3, E"
  :precision binary64

  :herbie-target
  (- (* 6.0 x) (* 9.0 (* x x)))

  (* (* 3.0 (- 2.0 (* x 3.0))) x))